Advertisement

Mathematical Geosciences

, Volume 48, Issue 4, pp 463–485 | Cite as

A Class-Kriging Predictor for Functional Compositions with Application to Particle-Size Curves in Heterogeneous Aquifers

  • Alessandra Menafoglio
  • Piercesare Secchi
  • Alberto Guadagnini
Article

Abstract

This work addresses the problem of characterizing the spatial field of soil particle-size distributions within a heterogeneous aquifer system. The medium is conceptualized as a composite system, characterized by spatially varying soil textural properties associated with diverse geomaterials. The heterogeneity of the system is modeled through an original hierarchical model for particle-size distributions that are here interpreted as points in the Bayes space of functional compositions. This theoretical framework allows performing spatial prediction of functional compositions through a functional compositional Class-Kriging predictor. To tackle the problem of lack of information arising when the spatial arrangement of soil types is unobserved, a novel clustering method is proposed, allowing to infer a grouping structure from sampled particle-size distributions. The proposed methodology enables one to project the complete information content embedded in the set of heterogeneous particle-size distributions to unsampled locations in the system. These developments are tested on a field application relying on a set of particle-size data observed within an alluvial aquifer in the Neckar river valley, in Germany.

Keywords

Geostatistics Functional compositions Clustering  Particle-size curves Groundwater Hydrogeology 

Notes

Acknowledgments

Financial support of MIUR (Project “Innovative methods for water resources under hydro-climatic uncertainty scenarios”, PRIN 2010/2011) is gratefully acknowledged. Support from the European Union’s Horizon 2020 Research and Innovation programme (Project “Furthering the knowledge Base for Reducing the Environmental Footprint of Shale Gas Development” FRACRISK—Grant Agreement No. 640979) is also acknowledged.

References

  1. Aitchison J (1982) The statistical analysis of compositional data. J R Stat Soc B 44(2):139–177Google Scholar
  2. Aitchison J (1986) The statistical analysis of compositional data. Chapman and Hall, LondonGoogle Scholar
  3. Cressie N (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  4. Delicado P (2011) Dimensionality reduction when data are density functions. Comput Stat Data Anal 55(1):401–420CrossRefGoogle Scholar
  5. Egozcue JJ (2009) Reply to “On the Harker Variation Diagrams; ...” by J.A. Cortés. Math Geosci 41(7): 829-834Google Scholar
  6. Egozcue JJ, Díaz-Barrero JL, Pawlowsky-Glahn V (2006) Hilbert space of probability density functions based on aitchison geometry. Acta Math Sin 22(4):1175-1182Google Scholar
  7. Egozcue JJ, Pawlowsky-Glahn V, Tolosana-Delgado R, Ortego M, van den Boogaart K (2013) Bayes spaces: use of improper distributions and exponential families. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales Serie A Matematicas 107(2):475-486Google Scholar
  8. Franco-Villoria M, Ignaccolo R (2014) Uncertainty evaluation in functional kriging with external drift. In: Bongiorno EG, Salinelli E, Goia A, Vieu P (eds) Contributions in infinite-dimensional statistics and related topics. Esculapio Pub Co., Bologna, pp 191–196Google Scholar
  9. Fréchet M (1948) Les éléments Aléatoires de Nature Quelconque dans une Espace Distancié. Annales de L’Institut Henri Poincaré 10(4):215–308Google Scholar
  10. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer series in statistics. Springer, BerlinCrossRefGoogle Scholar
  11. Hron K, Menafoglio A, Templ M, Hruzova K, Filzmoser P (2015) Simplicial principal component analysis for density functions in Bayes spaces. Comput Stat Data Anal. doi: 10.1016/j.csda.2015.07.007
  12. Marron JS, Alonso AM (2014) Overview of object oriented data analysis. Biom J 56(5):732–753CrossRefGoogle Scholar
  13. Martìn MA, Rey JM, Taguas FJ (2005) An entropy-based heterogeneity index for mass-size distributions in earth science. Ecol Model 182:221–228CrossRefGoogle Scholar
  14. McQueen J (1967) Some methods for classification and analysis of multivariate observations. 5th Berkeley symposium on mathematics. Statistics and probability, vol 1, pp 281-298Google Scholar
  15. Menafoglio A, Guadagnini A, Secchi P (2014) A Kriging approach based on Aitchison geometry for the characterization of particle-size curves in heterogeneous aquifers. Stoch Environ Res Risk Assess 28(7):1835–1851CrossRefGoogle Scholar
  16. Menafoglio A, Secchi P, Dalla Rosa M (2013) A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space. Electron J Stat 7:2209–2240CrossRefGoogle Scholar
  17. Nerini D, Ghattas B (2007) Classifying densities using functional regression trees: applications in oceanology. Comput Stat Data Anal 51(10):4984–4993CrossRefGoogle Scholar
  18. Pawlowsky-Glahn V, Buccianti A (2011) Compositional data analysis. Theory and applications. Wiley, New YorkGoogle Scholar
  19. Pawlowsky-Glahn V, Egozcue JJ (2001) Geometric approach to statistical analysis in the symplex. Stoch Environ Res Risk Assess 15:384–398CrossRefGoogle Scholar
  20. Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015) Modeling and analysis of compositional data. Statistics in practice. Wiley, New YorkGoogle Scholar
  21. Pigoli D, Menafoglio A, Secchi P (2013) Kriging prediction for manifold-valued random field. CRiSM Paper No. 13-18, University of WarwickGoogle Scholar
  22. Ramsay J, Silverman B (2005) Functional data analysis, 2nd edn. Springer, New YorkGoogle Scholar
  23. Riva M, Guadagnini A, Fernández-García D, Sánchez-Vila X, Ptak T (2008) Relative importance of geostatistical and transport models in describing heavily tailed breakthrough curves at the Lauswiesen site. J Contam Hydrol 101:1–13CrossRefGoogle Scholar
  24. Riva M, Guadagnini L, Guadagnini A (2010) Effects of uncertainty of lithofacies, conductivity and porosity distributions on stochastic interpretations of a field scale tracer test. Stoch Environ Res Risk Assess 24:955–970. doi: 10.1007/s00477-010-0399-7 CrossRefGoogle Scholar
  25. Riva M, Guadagnini L, Guadagnini A, Ptak T, Martac E (2006) Probabilistic study of well capture zones distributions at the Lauswiesen field site. J Contam Hydrol 88:92–118CrossRefGoogle Scholar
  26. Riva M, Sánchez-Vila X, Guadagnini A (2014) Estimation of spatial covariance of log-conductivity from particle-size data. Water Resour Res 50(6):5298–5308. doi: 10.1002/2014WR015566 CrossRefGoogle Scholar
  27. Sangalli LM, Secchi P, Vantini S (2014) Object oriented data analysis: a few methodological challenges. Biom J 56(5):774–777CrossRefGoogle Scholar
  28. Tolosana-Delgado R, Pawlowsky-Glahn V, Egozcue JJ (2008a) Indicator kriging without order relation violations. Math Geosci 40(3):327–347CrossRefGoogle Scholar
  29. Tolosana-Delgado R, Pawlowsky-Glahn V, Egozcue JJ (2008b) Simplicial indicator kriging. J China Univ Geosci 19(1):65–71CrossRefGoogle Scholar
  30. Tolosana-Delgado R, van den Boogaart KG, Pawlowsky-Glahn V (2011) Geostatistics for compositions. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, Chichester. doi: 10.1002/9781119976462.ch6 Google Scholar
  31. van den Boogaart K, Egozcue JJ, Pawlowsky-Glahn V (2010) Bayes linear spaces. SORT 34(2):201–222Google Scholar
  32. van den Boogaart KG, Egozcue JJ, Pawlowsky-Glahn V (2014) Bayes Hilbert spaces. Aust NZ J Stat 56:171–194CrossRefGoogle Scholar
  33. Vukovic M, Soro A (1992) Determination of hydraulic conductivity of porous media from grain-size composition. Water Resources Publications, LittletonGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2015

Authors and Affiliations

  • Alessandra Menafoglio
    • 1
  • Piercesare Secchi
    • 1
  • Alberto Guadagnini
    • 2
    • 3
  1. 1.MOX-Department of MathematicsPolitecnico di MilanoMilanoItaly
  2. 2.Dipartimento di Ingegneria Civile e AmbientalePolitecnico di MilanoMilanoItaly
  3. 3.Department of Hydrology and Water ResourcesThe University of ArizonaTucsonUSA

Personalised recommendations